Annali di Matematica Pura ed Applicata

, Volume 136, Issue 1, pp 183–212 | Cite as

Interpolation of cosine operator functions

  • Ronald H. W. Hoppe


Using basic techniques from the theory of interpolation spaces equivalence theorems are established for the intermediate spaces between a given Banach space A and the domain D(Λr) of the r-th power of the infinitesimal generator Λ of a strongly continuous cosine operator function C. The results are applied to the study of second order evolution equations including regularity, order reduction and approximation by finite difference methods.


Banach Space Finite Difference Evolution Equation Difference Method Operator Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. A. Baker -V. A. Dougalis -S. M. Serbin,An approximate theorem for second-order evolution equations, Numer. Math.,35 (1980), pp. 127–142.Google Scholar
  2. [2]
    G. A. Baker -V. A. Dougalis -S. M. Serbin,High order accurate two-step approximations for hyperbolic equations, R.A.I.R.O. Analyse Numérique,13 (1979), pp. 201–226.Google Scholar
  3. [3]
    H. Berens,Interpolationsmethoden zur Banachlung von Approximationsprozessen auf Banachräumen, Lect. Notes in Math.,64, Springer, Berlin-Heidelberg-New York, 1968.Google Scholar
  4. [4]
    H. Berens -P. L. Butzer -U. Westphal,Representation of fractional powers of infinitesimal generators of semigroups, Bull. Amer. Math. Soc.,74 (1968), pp. 191–196.Google Scholar
  5. [5]
    J. Berg -J. Löfström,Interpolation spaces, Springer, Berlin-Heidelberg-New York, 1976.Google Scholar
  6. [6]
    P. Brenner -V. Thomée -L. B. Wahlbin,Besov spaces and applications to difference methods for initial value problems, Lect. Notes in Math.,434, Springer, Berlin-Heidelberg-New York, 1975.Google Scholar
  7. [7]
    P. L. Butzer -H. Berens,Semigroups of operators and approximation, Springer, Berlin-Heidelberg-New York, 1967.Google Scholar
  8. [8]
    R. W. Carroll,Transmutation and operator differential equations, North-Holland, Amsterdam-New York-Oxford, 1979.Google Scholar
  9. [9]
    G. Da Prato -E. Giusti,Una caratterizzazione dei generatori di funzioni coseno astratte, Boll. Un. Mat. Ital.,22 (1967), pp. 357–362.Google Scholar
  10. [10]
    J. Dettmann,The wave, Laplace, and heat equations and related transforms, Glasgow Math. J.,11 (1970), pp. 117–125.Google Scholar
  11. [11]
    H. O. Fattorini,Ordinary differential equations in linear topologieal spaces I, J. Diff. Equat.,5 (1968), pp. 72–105.Google Scholar
  12. [12]
    H. O. Fattorini,Ordinary differential equations in linear topological spaces II, J. Diff. Equat.,6 (1969), pp. 50–70.Google Scholar
  13. [13]
    R. D. Grigorieff,Numerik gewöhnlicher Differentialgleichungen 2, Teubner, Stuttgart, 1977.Google Scholar
  14. [14]
    R. H. W.Hoppe,Two-step methods generated by Turán type quadrature formulas in the approximate solution of evolution equations of hyperbolic type, to appear in Math. of Comp.Google Scholar
  15. [15]
    R. H. W.Hoppe,Representation of fractional powers of infinitesimal generators of cosine operator functions, Results in Mathematics,7 (1984) (in press).Google Scholar
  16. [16]
    J. L. Lions,Théorèms de traces et d'interpolation I-V, Ann. Scuola Norm. Sup. Pisa,13 (1959), pp. 389–404;ibid.,14 (1960), pp. 317–331; J. Math. Pures Appl.,42 (1963), pp. 195–203; Math. Ann.,151 (1963), pp. 42–56; An. Acad. Brasil Ci.,35 (1963), pp. 1–10.Google Scholar
  17. [17]
    W. Littmann,The wave operator and L p-norms, J. Math. Mech.,12 (1963), pp. 55–68.Google Scholar
  18. [18]
    L. A. Murarei,The Cauchy problem for the wave-equation in L p-norm, Trudy Mat. Inst. Steklova,53 (1968), pp. 172–180.Google Scholar
  19. [19]
    B. Nagy,On cosine operator functions in Banach spaces, Acta Sci. Math. Szeged,36 (1974), pp. 281–290.Google Scholar
  20. [20]
    J. Peetre,A theory of interpolation of normed spaces, Notas de matematica,39 (1968), pp. 1–86.Google Scholar
  21. [21]
    M. Sova,Cosine operator functions, Rozprawy Mat.,49 (1966), pp. 1–47.Google Scholar
  22. [22]
    H. Triebel,Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam-New York-Oxford, 1978.Google Scholar
  23. [23]
    U. Westphal,Ein Kalkül für gebrochene Potenzen infinitesimaler Erzeuger von Halbgruppen und Gruppen, Teil I: Halbgruppenerzeuger, Compositio Math.,22 (1970), pp. 67–103.Google Scholar
  24. [24]
    U. Westphal,Ein Kalkül für gebrochene Potenzen infinitesimaler Erzeuger von Halbgruppen und Gruppen, Teil II: Gruppenerzeuger, Compositio Math.,22 (1970), pp. 104–136.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1984

Authors and Affiliations

  • Ronald H. W. Hoppe
    • 1
  1. 1.BerlinWest Germany

Personalised recommendations